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Conclusion

    This book advocates for a platonist view of logic and mathematics, in order to truly understand the formal sciences’ relationship with natural sciences. Philosophers like Mario Bunge, Philip Kitcher, Imre Lakatos, Karl Popper, Hilary Putnam, W. V. O. Quine among many others have expressed deep dissatisfaction with the view that there can exist an ideal world or a fregean “third realm”.

    Throughout our analysis we have seen that many of the arguments presented against platonism are completely false, products of misunderstandings, and sometimes non sequiturs. Behind all of these arguments, there is only one prejudice: there is no possibility for an abstract logico-mathematical reality. If there is no ideal realm, mathematics and logic cannot provide true knowledge like science provides knowledge. In the best cases, some philosophers can accept a certain kind of formal knowledge. In the worst of cases, there is essentially no difference at all between the formal and the natural, all our theories are posits.

    Against the former, we recognize that it is possible that we make up concepts which may have foreseen or unforeseen logical consequences. But we can question if before the concept was invented everything that existed did not conform to the objective laws of logic and mathematics. Besides, the physical world and all of its laws did exist before us, and they are all subject to the same objectual relations. These logical-mathematical truths would be the condition of possibility of all beings and propositions.

    Against those who hold that there is no difference between posits of formal science and the posits of natural science, a qualitative difference between such posits must be recognized. To distinguish between logic and mathematics on one hand and natural science on the other, we can observe that in these two fields the epistemological foundations and knowing activities are very different. In that sense, it is useful to distinguish between them. It is also useful to distinguish between qualitative aspects of logic and mathematics such as axioms, theorems, and proofs which require no sensory input whatsoever, from the qualitative aspects of natural science, which use the laws of formal sciences to formulate theories, that are confirmed or refuted in light of experience.

    Hence, the outcome of this analysis with respect to the issue of the underdetermination of science is perfectly clear: science is unable to revise logic and mathematics. Of course, it is possible to argue that a posteriori phenomena can stimulate mathematical works, like in Newton's and Leibniz's cases. However, all that this shows is that there is a dialogue between formal science and natural science, but none can be reduced to the other, and one cannot be the servant of the other. Let each field have its own investigations, its own knowledge, and its own discovery of the truth.